Modeling synchronization in human musical rhythms using Impulse Pattern Formulation (IPF)
Simon Linke, Rolf Bader, Robert Mores · arXiv (Cornell University) · 2021
In the publication, supplemented by these sound examples, the Impulse Pattern Formulation is used to model the synchronization of musicians to a collective tempo. Several click tracks are numerically created, representing eighth notes played by a musician or a metronome for different tempo changes. By replacing every beat with a sound sample, audio files are created for a more musical evaluation of the results. The IPF is represented by a cowbell and the underlying click track with claves. Those sound files are in stereo, whereby the click track is at the left channel, and the IPF's signal is at the right channel. Thus, e.g., the balance potentiometer of a stereo system can be used to blend both sounds freely. The practical examples are: Fig.2: IPF reacts to four different step changes in tempo. Fig.5: The IPF reacting to the same changes in tempo as shown in Figure 2, when changing the tempo linear during 24 beats instead of step changes. Fig.8: IPF adapting to a noisy click track: the upper line (a) and b)) shows white noise, and the lower line (c) and d)) Brownian noise. On the left (a) and c)), the fluctuation is \(\pm1~\%\), and on the right (b) and d)) \(\pm 5~\%\). Fig.9: IPF adapting to a sinusoidally modulated click track: the upper line (a) and b)) shows a modulation period of 32 eighth notes, and the lower line (c) and d)) shows a modulation period of 8 eighth notes. On the left (a) and c)), the amplitude is 36 bpm, and on the right (b) and d)) 6 bpm, both centered around 113 bpm. Fig.12: Several scenarios shown in Figures 2, 5, 8, and 9 applied to an extended IPF which considers phase differences: a) step change from 120 to 100 bpm, b) linear change from 120 to 130 bpm, c) \(\pm 5~\%\) Brownian noise added to a 120 bpm click track and d) sinusoidal modulation with a period length of 32 eight notes varied \(\pm 6~bpm\) around 113 bpm. Fig.13: Several scenarios shown in Figures 2, 5, 8, and 9 applied to an extended IPF optimized for polyrhythms: a) step change from 120 to 140 bpm, b) linear change from 100 to 120 bpm, c) \(\pm 5~\%\) Brownian noise added to a 90 bpm click track and d) sinusoidal modulation with a period length of 32 eight notes varied \(\pm 6~bpm\) around 113 bpm. In all Figures, blue lines refer to the tempo of the click track, and red lines correspond to the tempo of the IPF. The single crosses represent single beats. A more in-depth description of how these sounds were synthesized can be found in the publication supplemented by these examples: Linke, S., Bader, R., & Mores, R. (2021). Modeling synchronization in human musical rhythms using Impulse Pattern Formulation (IPF). http://arxiv.org/pdf/2112.03218v1